The study of proper holomorphic maps between balls goes back to Poincaré [1]} and Alexander [2]} who showed that the proper holomorphic self-maps of a ball in \(\mathbb {C}^N\) , \(N\geqslant 2\) , are precisely the automorphisms which have explicit parametrizations. When the target is a ball of higher dimension, the discovery of inner functions implies that there are proper holomorphic maps between balls that do not extend to any boundary point [3]}, [4]}, [5]}, [6]}. On the other hand, by a well-known result of Forstnerič [7]}, proper holomorphic maps between balls, which extend sufficiently smooth to a boundary point, extend to rational maps with no poles on the boundary and induce global CR maps between spheres.
